On Strong Product of Two Fuzzy Graphs

Σχετικά έγγραφα
Fractional Colorings and Zykov Products of graphs

2 Composition. Invertible Mappings

Commutative Monoids in Intuitionistic Fuzzy Sets

Homomorphism in Intuitionistic Fuzzy Automata

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

( ) ( t) ( 0) ( ) dw w. = = β. Then the solution of (1.1) is easily found to. wt = t+ t. We generalize this to the following nonlinear differential

Finite Field Problems: Solutions

A Note on Intuitionistic Fuzzy. Equivalence Relation

C.S. 430 Assignment 6, Sample Solutions

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Statistical Inference I Locally most powerful tests

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in

Homomorphism of Intuitionistic Fuzzy Groups

Example Sheet 3 Solutions

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Cubic Γ-n normed linear spaces

Congruence Classes of Invertible Matrices of Order 3 over F 2

EE512: Error Control Coding

4.6 Autoregressive Moving Average Model ARMA(1,1)

Every set of first-order formulas is equivalent to an independent set

Homework 3 Solutions

Appendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

( ) ( ) ( ) Fourier series. ; m is an integer. r(t) is periodic (T>0), r(t+t) = r(t), t Fundamental period T 0 = smallest T. Fundamental frequency ω

Matrices and Determinants

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

Section 8.3 Trigonometric Equations

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Lecture 2. Soundness and completeness of propositional logic

ST5224: Advanced Statistical Theory II

Math221: HW# 1 solutions

A summation formula ramified with hypergeometric function and involving recurrence relation

Lecture 12 Modulation and Sampling

CRASH COURSE IN PRECALCULUS

Areas and Lengths in Polar Coordinates

Solution Series 9. i=1 x i and i=1 x i.

Strain gauge and rosettes

A Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

DIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS

1. Introduction and Preliminaries.

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER

New bounds for spherical two-distance sets and equiangular lines

GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)

MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)

Parametrized Surfaces

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

w o = R 1 p. (1) R = p =. = 1

Reminders: linear functions

Mean-Variance Analysis

Homework 8 Model Solution Section

Second Order Partial Differential Equations

Problem Set 3: Solutions

Solutions to Exercise Sheet 5

derivation of the Laplacian from rectangular to spherical coordinates

Srednicki Chapter 55

Other Test Constructions: Likelihood Ratio & Bayes Tests

Areas and Lengths in Polar Coordinates

Uniform Convergence of Fourier Series Michael Taylor

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

Overview. Transition Semantics. Configurations and the transition relation. Executions and computation

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

The challenges of non-stable predicates

Tridiagonal matrices. Gérard MEURANT. October, 2008

On a four-dimensional hyperbolic manifold with finite volume

Managing Production-Inventory Systems with Scarce Resources

Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee

Cyclic or elementary abelian Covers of K 4

Homomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PS-algebras

Linear singular perturbations of hyperbolic-parabolic type

Abstract Storage Devices

Intuitionistic Fuzzy Ideals of Near Rings

Right Rear Door. Let's now finish the door hinge saga with the right rear door

Partial Differential Equations in Biology The boundary element method. March 26, 2013

A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings

Inverse trigonometric functions & General Solution of Trigonometric Equations

Approximation of distance between locations on earth given by latitude and longitude

SOME PROPERTIES OF FUZZY REAL NUMBERS

The Simply Typed Lambda Calculus

Necessary and sufficient conditions for oscillation of first order nonlinear neutral differential equations

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

9.1 Introduction 9.2 Lags in the Error Term: Autocorrelation 9.3 Estimating an AR(1) Error Model 9.4 Testing for Autocorrelation 9.

F19MC2 Solutions 9 Complex Analysis

Trigonometric Formula Sheet

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.

Lecture 13 - Root Space Decomposition II

5. Choice under Uncertainty

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Generating Set of the Complete Semigroups of Binary Relations

Bounding Nonsplitting Enumeration Degrees

Section 7.6 Double and Half Angle Formulas

ω = radians per sec, t = 3 sec

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Math 6 SL Probability Distributions Practice Test Mark Scheme

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

Transcript:

Inernaional Journal of Scienific and Research Publicaions, Volume 4, Issue 10, Ocober 014 1 ISSN 50-3153 On Srong Produc of Two Fuzzy Graphs Dr. K. Radha* Mr.S. Arumugam** * P.G & Research Deparmen of Mahemaics, Periyar E.V.R. College, Tiruchirapalli-6003 ** Gov. High School, Thinnanur, Tiruchirapalli-61006. Absrac- In his paper, he srong produc of wo fuzzy graphs is defined. I is proved ha when wo fuzzy graphs are effecive hen heir srong produc is always effecive and i is proved ha he srong produc of wo complee fuzzy graphs is complee. Also i is proved ha he srong produc of wo conneced fuzzy graphs is always conneced. The lower and upper runcaions of he srong produc of wo fuzzy graphs are obained. The degree of a verex in he srong produc of wo fuzzy graphs is obained. A relaionship beween he direc sum and he srong produc of wo fuzzy graphs is obained. Index Terms- Fuzzy Graph, Direc Sum, Srong Produc, Effecive Fuzzy Graph, Connecedness, Upper and Lower Truncaions. 010 Mahemaics Subjec Classificaion: 03E7, 05C07. F I. INTRODUCTION uzzy graph heory was inroduced by Azriel Rosenfeld in 1975. The properies of fuzzy graphs have been sudied by Azriel Rosenfeld[9]. Laer on, Bhaacharya[7] gave some remarks on fuzzy graphs, and some operaions on fuzzy graphs were inroduced by Mordeson.J.N. and Peng.C.S.[3]. The conjuncion of wo fuzzy graphs was defined by Nagoor Gani.A and Radha.K.[4]. We defined he direc sum of wo fuzzy graphs and sudied he effeciveness, connecedness and regular properies of he direc sum of wo fuzzy graphs [8]. In his paper, he srong produc of wo fuzzy graphs is defined. I is proved ha when wo fuzzy graphs are effecive hen heir srong produc is always effecive and i is proved ha he srong produc of wo complee fuzzy graphs is complee. Also i is proved ha he srong produc of wo conneced fuzzy graphs is always conneced. The lower and upper runcaions of he srong produc of wo fuzzy graphs are obained. The degree of a verex in he srong produc of wo fuzzy graphs is obained. A relaionship beween he direc sum and he srong produc of wo fuzzy graphs is obained. Firs le us recall some preliminary definiions ha can be found in [1]-[9]. A fuzzy graph G is a pair of funcions (σ, μ) where σ is a fuzzy subse of a non empy se V and μ is a symmeric fuzzy relaion on σ. The underlying crisp graph of G:(σ, μ) is denoed by G*(V, E) where E V V. Le G:(σ, μ) be a fuzzy graph. The underlying crisp graph of G:(σ, μ) is denoed by G*:(V, E) where E V V. A fuzzy graph G is an effecive fuzzy graph if μ(u,v) = σ(u) σ(v) for all (u,v)e and G is a complee fuzzy graph if μ(u,v) = σ(u) σ(v) for all u,vv. Therefore G is a complee fuzzy graph if and only if G is an effecive fuzzy graph and G* is complee. (σ, μ ) is a spanning fuzzy subgraph of (σ,μ) if σ =σ and μ μ, ha is, if σ (u) = σ (u) for every uv and μ (e) μ(e) for every ee.. The degree of a verex u of a fuzzy graph G is defined as d G (u) (uv) (uv) uv uve The Caresian produc of wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is defined as a fuzzy graph G= G 1 G : (σ 1 σ, μ 1 μ ) on G*:(V,E) where V = V 1 V and E = {((u 1, v 1 )(u, v )) / u 1 = u, v 1 v E or v 1 = v,u 1 u E 1 } wih (σ 1 σ )(u,v) = σ 1 (u) σ (v), for all (u, v) V 1 V and 1 (u 1) (v1v ),if u1 u, v1v E ( 1 ) u 1, v1 u, v (v 1) 1(u1 u ),if v 1 v,u1u E1 The conjuncion or he ensor produc of wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is defined as a fuzzy graph G = G 1 G : (σ, μ) on G*:(V,E) where V = V 1 V and E = {((u 1, v 1 )(u, v )) / u 1 u E 1, v 1 v E } wih σ(u 1, v 1 ) = σ 1 (u 1 ) σ (v 1 ), for u,v u,v (u u ) (v v ),for all u,v u,v E. all (u 1, v 1 ) V 1 V and 1 1 1 1 1 1 1 If G 1 :(σ 1,μ 1 ) and G :(σ,μ ) are wo fuzzy graphs such ha σ 1 μ hen σ μ 1 [6]. The lower and upper runcaions of σ a level, 0 < 1, are he fuzzy subses σ () and σ () defined respecively by, (u),if u,if u (u) and (u) 0,if u (u),if u.

Inernaional Journal of Scienific and Research Publicaions ISSN 50-3153 Le G:(σ,μ) be a fuzzy graph wih underlying crisp graph G*:(V,E). Take V () = σ, E () = μ. Then G () :(σ (),μ () ) is a fuzzy graph wih underlying crisp graph G () *:(V (), E () ). This is called he lower runcaion of he fuzzy graph G a level. Here V () and E () may be proper subses of V and E respecively. Take V () = V, E () = E. Then G () :(σ (), μ () ) is a fuzzy graph wih underlying crisp graph G () *:(V (),E () ). This is called he upper runcaion of he fuzzy graph G a level [5]. Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) denoe wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Le V = V 1 V and le E = {uv / u,vv; uv E 1 or uv E bu no boh }. Define G:(σ, μ) by (u),if u V V (uv),if uv E (u) (u),if u V V and (uv) 1 1 1 1 1 (uv),if uv E 1(u) (u),if u V1 V Then if uv E 1, μ(uv) = μ 1 (uv) σ 1 (u) σ 1 (v) σ(u) σ(v), if uv E, μ(uv) = μ (uv) σ (u) σ (v) σ(u) σ(v). Therefore (σ, μ) defines a fuzzy graph. This is called he direc sum of wo fuzzy graphs. II STRONG PRODUCT Definiion.1 Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) denoe wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. The normal produc of G 1 * and G * is G* = G 1 * G *: (V, E) where V = V 1 V and E = {(u 1, v 1 )(u, v ) / u 1 = u, v 1 v E or v 1 =v,u 1 u E 1 or u 1 u E 1 and v 1 v E }. Define G:(σ, μ), where σ = σ 1 σ and μ = μ 1 μ by σ(u 1, v 1 ) = σ 1 (u 1 ) σ (v 1 ), for all (u 1, v 1 ) V 1 V and 1 (u 1) (v1v ),if u1 u, v1v E u 1, v1 u, v (v 1) 1(u1 u ),if v1 v,u1u E1 1(u1u ) (v1v ),if u1u E 1, v1v E If u 1 = u, v 1 v E, σ 1 (u 1 ) μ (v 1 v ) = σ 1 (u 1 ) σ 1 (u ) μ (v 1 v ) σ 1 (u 1 ) σ 1 (u ) σ (v 1 ) σ (v ) = σ 1 (u 1 ) σ (v 1 ) σ 1 (u ) σ (v ) = σ(u 1, v 1 ) σ(u, v ) Similarly if v 1 = v, u 1 u E 1, σ (v 1 ) μ 1 (u 1 u ) σ(u 1, v 1 ) σ(u, v ) If u 1 u E 1 and v 1 v E, μ 1 (u 1 u ) μ (v 1 v ) σ 1 (u 1 ) σ 1 (u ) σ (v 1 ) σ (v ) = σ(u 1, v 1 ) σ(u, v ) Hence μ((u 1, v 1 )(u, v )) σ(u 1, v 1 ) σ(u, v ). Therefore G:(σ, μ) is a fuzzy graph. This is called he normal produc or he srong produc of he fuzzy graphs G 1 and G and is denoed by G 1 G. Example. The following Figure1 gives an example of he srong produc of wo fuzzy graphs. Figure 1: The srong produc G 1 G of G 1 and G Theorem.3: If G 1 and G are wo effecive fuzzy graphs, hen G 1 G is an effecive fuzzy graph. Le G 1 and G be effecive fuzzy graphs. Then μ 1 (u 1 u ) = σ 1 (u 1 )σ 1 (u ) for any u 1 u E 1 and μ (v 1 v ) = σ (v 1 ) σ (v ) for any v 1 v E. Therefore proceeding as in he definiion, If u 1 = u, v 1 v E, μ((u 1, v 1 )(u, v )) = σ 1 (u 1 ) μ (v 1 v ) = σ 1 (u 1 )σ 1 (u ) σ (v 1 )σ (v ) = (σ 1 (u 1 )σ 1 (u )) (σ (v 1 )σ (v )) = σ(u 1, v 1 ) σ(u, v ). Similarly, If v 1 = v, u 1 u E 1, μ((u 1, v 1 )(u, v )) = σ(u 1, v 1 ) σ(u, v ) If u 1 u E 1 and v 1 v E, μ((u 1, v 1 )(u, v )) = σ(u 1, v 1 ) σ(u, v ).

Inernaional Journal of Scienific and Research Publicaions 3 ISSN 50-3153 Hence G 1 G is an effecive fuzzy graph. Theorem.4: If G 1 and G are wo complee fuzzy graphs, hen G 1 G is a complee fuzzy graph. Le G 1 and G be complee fuzzy graphs. Then G 1 and G are effecive fuzzy graphs and G 1 * andg * are complee graphs. Therefore G 1 G is an effecive fuzzy graph by Theorem. and G 1 * G * is a complee graph. Hence G 1 G is a complee fuzzy graph. Example.5: The following Figure gives an example of he srong produc of wo effecive fuzzy graphs. Figure : The srong produc G 1 G of wo effecive fuzzy graphs G 1 and G Example.6: The following Figure 3 gives an example of he srong produc of wo complee fuzzy graphs. Figure 3: The srong produc G 1 G of wo complee fuzzy graphs G 1 and G Theorem.7: The srong produc of wo conneced fuzzy graphs is always a conneced fuzzy graph. Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) be wo conneced fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Le V 1 = {u 1, u,..,u m } and V = {v 1, v,..,v n }. The srong produc of wo conneced fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) can be aken as G:(σ, μ) where σ = σ 1 σ and μ = μ 1 μ. Now consider he m sub graphs of G wih he verex ses {u i v 1, u i v,..,u i v n } for i=1,,,m. Each of hese sub graphs of G is conneced since he u i s are he same and since G is conneced, each v i is adjacen o a leas one of he verices in V. Also since G 1 is conneced, each u i is adjacen o a leas one of he verices in V 1. Therefore here exiss a leas one edge beween any pair of he above m sub graphs. Hence G is a conneced fuzzy graph.

Inernaional Journal of Scienific and Research Publicaions 4 ISSN 50-3153 III TRUNCATIONS OF THE STRONG PRODUCT OF TWO FUZZY GRAPHS Theorem 3.1: (G 1 G ) () = G 1() G () and (G 1 G ) () = G () 1 G (). 1 u, v,if 1 u, v We have 1 u, v () 0,if 1 u, v Now (σ 1() σ () ) (u,v) = σ 1() (u) σ () (v) If (σ 1 σ )(u, v), hen σ 1 (u) σ (v) σ 1 (u) and σ (v) σ 1() (u) = σ 1 (u), σ () (v) = σ (v) σ 1() (u) σ () (v) = σ 1 (u)σ (v)= (σ 1 σ )(u, v). If (σ 1 σ )(u, v) <, hen σ 1 (u) σ (v) < eiher σ 1 (u) <, σ (v) or σ 1 (u), σ (v) < or σ 1 (u) <, σ (v) < σ 1() (u) =0, σ () (v) = σ (v) or σ 1() (u) = σ 1 (u), σ () (v) = 0 or σ 1() (u) = 0, σ () (v) = 0 σ 1() (u)σ () (v) = 0. 1 u, v,if 1 u, v Therefore 1() () u, v 1() u () v 0,if 1 u, v Hence (σ 1 σ ) () (u,v) = (σ 1() σ () ) (u,v) for every (u,v) V 1 V. Now 1 u 1, v1 u, v 1 1 1 1 1 1 1 1 1 u, v u, v,if u, v u, v () 0,if u, v u, v If (σ 1 σ )(u, v), hen σ 1 (u 1 ) μ (v 1 v ) or σ (v 1 ) μ 1 (u 1 u ) or μ 1 (u 1 u ) μ (v 1 v ) Proceeding as above, we can show ha 1 u1 v 1, v,if 1 u1 v 1, v u1 1 u 1,u,if v1 1 u 1,u 1() () u 1, v1 u, v 1 u 1,u v 1, v,if 1 u 1,u v 1, v 0,if 1 u 1,u v 1, v 1 1 1 1 1 1 1 1 1 u, v u, v,if u, v u, v 0,if u, v u, v Therefore (μ 1 μ ) () ( (u 1, v 1 )(u, v )) = (μ 1() μ () )((u 1, v 1 )(u, v )) for every edge (u 1,v 1 )(u, v ) in G 1 G. Hence (G 1 G ) () = G 1() G (). Proceeding in he same way, we can show ha (G 1 G ) () = G () 1 G (). IV. DEGREE OF A VERTEX IN THE STRONG PRODUCT OF TWO FUZZY GRAPHS The degree of any verex in he srong produc G 1 G of wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is given by, d G G (u i, v j) 1 (u i ) (v jv ) 1(u iu k ) 1(v j) 1(u iu k ) (v jv ). This expression can be 1 ui u k,vjve uiuke 1,v j = v uiuke 1,v jve simplified using he erms of he degrees of verices in G 1 and G wih some consrains. Theorem 4.1: If G 1 :(σ 1,μ 1 ) and G :(σ,μ ) are wo fuzzy graphs such ha σ 1 μ and σ μ 1 and μ 1 μ = c (a consan), hen he degree of a verex in he srong produc of he wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is given by, d (u,v ) d (v ) d (u ) [d (u )d (v )]c. G * * 1G i j G j G1 i G i j 1 G Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) be wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Suppose ha σ 1 μ and σ μ 1 and μ 1 μ = c (a consan), hen and 1 1 1 1 1 d (u, v ) (u ) (v v ) (u u ) (v ) (u u ) (v v ) G1G i j 1 i j 1 i k j 1 i k j. ui u k,vjve uiuke 1,v j = v uiuke 1,v jve

Inernaional Journal of Scienific and Research Publicaions 5 ISSN 50-3153 d (u, v ) (v v ) (u u ) c G1G i j j 1 i k ui u k,vjve uiuke 1,v j = v uiuk E 1,v jve d (v ) d (u ) [d (u )d (v )]c. G * * j G1 i G i j 1 G Theorem 4.: If G 1 :(σ 1,μ 1 ) and G :(σ,μ ) are wo fuzzy graphs such ha σ 1 μ and σ μ 1 and μ 1 μ = C (a consan), hen he degree of a verex in he srong produc of he wo fuzzy graphs G 1 :(σ 1,μ 1 ) and G :(σ,μ ) is given by, d (u,v ) [1 d (v )]d (u ) [1 d (u )]d (v ) [d (u )d (v )]C. G * * * * 1G i j G j G 1 i G i G 1 j G i j 1 G Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) be wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Suppose ha σ 1 μ and σ μ 1 and μ 1 μ = C (a consan), hen and 1 1 1 1 1 d (u, v ) (u ) (v v ) (u u ) (v ) (u u ) (v v ) G1G i j 1 i j 1 i k j 1 i k j ui u k,v jve uiuk E 1,v j = v uiuk E 1,v jve (v v ) (u u ) [ (u u ) (v v ) (u u ) (v v )] j 1 i k 1 i k j 1 i k j ui u k,v jve uiuk E 1,v j = v uiuk E1,v jve d (v ) d (u ) (u u ) (v v ) [ (u u ) (v v )] d (v ) d (u ) d (v )d (u ) d (u )d (v ) C G * * j G1 i G j G 1 i G i G 1 j * G j G1 i uiuk E 1,v jve [1 d (v )]d (u ) [1 d (u )]d (v ) [d (u )d (v )]C. G j G1 i 1 i k j 1 i k j uiuk E 1,v jve uiuk E 1,v jve uiuk E 1,v jve * * * G i G 1 j G i j 1 G Theorem 4.3: If G 1 :(σ 1,μ 1 ) and G :(σ,μ ) are wo fuzzy graphs such ha σ 1 µ and μ 1 μ = c (a consan), hen he degree of a verex in he srong produc is given by, d (u,v ) d (v ) (u ) d (u ) [d (u )d (v )]c. G * * * 1G i j G j 1 i G1 i G i j 1 G Le G 1 :(σ 1,μ 1 ) and G :(σ,μ ) be wo fuzzy graphs wih underlying crisp graphs G 1 *:(V 1,E 1 ) and G *:(V,E ) respecively. Suppose ha σ 1 μ. Then σ μ 1. This implies ha σ 1 σ. Also μ 1 μ = c (a consan). d (u, v ) (u ) (v v ) (u u ) (v ) (u u ) (v v ) G1G i j 1 i j 1 i k j 1 i k j ui u k,v jve uiuk E 1,v j = v uiuk E 1,v jve (u ) (u u ) (u u ) (v v ) d (v ) (u ) d (u ) [d (u )d (v )]c. * G 1 i 1 i k 1 i k j ui u k,vjve uiuk E 1,v j = v uiuk E 1,v jve j 1 i G * * 1 i G i j 1 G Example 4.4: If G 1 :(σ 1, μ 1 ) and G :(σ, μ ) are wo fuzzy graphs such ha σ 1 μ and μ 1 μ = c (a consan), hen heir srong produc G G : (, ) is given in he following example. 1 Figure 4: The Srong Produc of wo fuzzy graphs such ha σ 1 μ and μ 1 μ =0.3. d (u,v ) d (v ) (u ) d (u ) [d (u )d (v )]c 10.3 0.3 110.3 0.9 G * * * 1G 1 1 G 1 1 1 G1 1 G 1 1 1 G

Inernaional Journal of Scienific and Research Publicaions 6 ISSN 50-3153 d (u,v ) d (v ) (u ) d (u ) [d (u )d (v )]c 0.4 0.3 10.3 1.7 G * * * 1G G 1 G1 G 1 G V. RELATIONSHIP BETWEEN THE DIRECT SUM AND THE STRONG PRODUCT Theorem 5.1: The srong produc of wo fuzzy graphs G 1 and G is he direc sum of he Caresian produc of G 1 and G and he conjuncion of G 1 and G. From he definiions, (σ 1 σ )(u,v) = (σ 1 σ )(u,v) = (σ 1 σ )(u,v) = σ 1 (u) σ (v) for every (u,v) V 1 V. So ((σ 1 σ ) (σ 1 σ ))(u, v) = σ 1 (u) σ (v) for every (u,v) V 1 V. Hence (σ 1 σ )(u,v) = ((σ 1 σ ) ( σ 1 σ ))(u, v) for every (u,v) V 1 V. From he definiions of Caresian produc and he conjuncion, 1 u1 v 1, v,if u1 u, v1v E 1 1 u 1, v1 u, v u1 1 u 1,u,if v1 v,u1u E1 1 u 1,u v1, v,if u1u E1 and v1v E Hence G 1 G = (G 1 G ) (G 1 G ). 1 u 1,v1 u,v VI. CONCLUSION In his paper, he srong produc of wo fuzzy graphs is defined. I is proved ha when wo fuzzy graphs are effecive hen heir srong produc is always effecive and i is proved ha he srong produc of wo complee fuzzy graphs is complee. Also i is proved ha he srong produc of wo conneced fuzzy graphs is always conneced. The lower and upper runcaions of he srong produc of wo fuzzy graphs are obained. The degree of a verex in he srong produc of wo fuzzy graphs is obained. A relaionship beween he direc sum and he srong produc of wo fuzzy graphs is obained. Operaion on fuzzy graph is a grea ool o consider large fuzzy graph as a combinaion of small fuzzy graphs and o derive is properies from hose of he small ones. Through his paper, a sep in ha direcion is made. REFERENCES [1] Frank Harary, Graph Thoery, Narosa / Addison Wesley, Indian Suden Ediion, 1988. [] John N. Modeson and Premchand S.Nair, Fuzzy Graphs and Fuzzy Hypergraphs, Physica-verlag Heidelberg, 000. [3] J.N.Mordeson and C.S. Peng, Operaions on fuzzy graphs, Informaion Sciences 79 (1994), 159-170. [4] Nagoorgani. A and Radha. K, Conjuncion of Two Fuzzy Graphs, Inernaional Review of Fuzzy Mahemaics, 008, Vol. 3, 95-105. [5] Nagoorgani. A and Radha. K, Some Properies of Truncaions of Fuzzy Graphs, Advances in Fuzzy Ses and Sysems, 009, Vol.4, No., 15-7. [6] Nagoorgani. A and Radha. K, Regular Propery of Fuzzy Graphs, Bullein of Pure and Applied Sciences, Vol.7E ( No.)008, 411-419. [7] P. Bhaacharya, Some Remarks on Fuzzy Graphs, Paern Recogniion Leer 6 (1987), 97-30. [8] Radha.K and Arumugam. S, On Direc Sum of Two Fuzzy Graphs, Inernaional Journal of Scienific and Research Publicaions, Volume 3, Issue 5, May 013, ISSN 50-3153. [9] Rosenfeld. A, (1975) "Fuzzy graphs". In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.), Fuzzy Ses and heir Applicaions o Cogniive and Decision Processes, Academic Press, New York, ISBN 97801775600, pp.77 95. AUTHORS Firs Auhor Dr. K. Radha, M.Sc.,M.Phil.,Ph.D., P.G & Research Deparmen of Mahemaics, Periyar E.V.R. College, Tiruchirapalli-6003. E-mail: radhagac@yahoo.com Second Auhor Mr.S. Arumugam, M.Sc.,M.Phil.,B.Ed.,(Ph.D.), Gov. High School, Thinnanur, Tiruchirapalli-61006. E-mail: anbu.saam@gmail.com Correspondence Auhor Dr. K. Radha, M.Sc.,M.Phil.,Ph.D., P.G & Research Deparmen of Mahemaics, Periyar E.V.R. College, Tiruchirapalli-6003. E-mail: radhagac@yahoo.com

2025 © DocPlayer.gr Πολιτική Απορρήτου | Όροι Χρήσης | Σχόλια